Question: $\sum\limits_{k=1}^{275 }{{(-5k + 12)}}=$
Answer: What is the question asking for? The question is asking for the sum of the values of $-5k + 12$ from $k = 1$ to $k = 275 $ : $(-5 \cdot 1 + 12) + (-5 \cdot 2 + 12) +... + (-5\cdot {275} +12)$ The series is arithmetic because the formula $-5k + 12$ is a linear function of $k$. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The number of terms $(n = {275})$ is the upper limit of the sigma notation. We need to find $a_1$ (the first term) and $a_{275}$ (the last term). Step 1: Find $a_1$ and $a_{275}$ (the first and the last term) $a_1 = -5(1) + 12 = {7}$ $a_{275} = -5(275) + 12 = {-1363}$ Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{275}}&= \dfrac {\left({7} + ({-1363}) \right)}{2} \cdot {275} \\\\ S_{{275}} &= -678 \left(275\right) \\\\ S_{{275}} &= -186{,}450\end{aligned}$ The answer $ -186{,}450 $